The equivalence of two discretenesses of triangulated categories

Abstract

Given an ST-triple $(\mathcal{C},\mathcal{D},M)$ one can associate a co-$t$-structure on $\mathcal{C}$ and a $t$-structure on $\mathcal{D}$. It is shown that the discreteness of $\mathcal{C}$ with respect to the co-$t$-structure is equivalent to the discreteness of $\mathcal{D}$ with respect to the $t$-structure. As a special case, the discreteness of $\mathcal{D}^b(\mathrm{mod} A)$ in the sense of Vossieck is equivalent to the discreteness of $K^b(\mathrm{proj} A)$ in a dual sense, where $A$ is a finite-dimensional algebra.